Cohomological equations of Riemannian flows and the Anosov diffeomorphism (Q2468455)

On an oriented connected \(C^\infty\) manifold M endowed with a diffeomorphism \(\gamma\), let \(g\in C^\infty (M)\) and \(X \in \Gamma^{\infty} (M)\). If there exists \(f\in C^{\infty} (M)\) such that \(f-f\circ\gamma = g\) (resp. \(Xf =g\)), then \(f\) is called a solution of the above discrete (resp. continuous) cohomological equation DCE (resp. CCE). The leafwise cohomology of a complete Riemannian Diophantine flow is computed here. Solving the DCE of \((M,\gamma)\) is equivalent to solve the CCE of the manifold endowed with the vector field obtained by the suspension of \((M,\gamma)\). The authors solve explicitly the DCE for the Anosov diffeomorphism on the torus \(\mathbb T^n\) defined by a hyperbolic and diagonalizable matrix \(A\in \text{SL}(n,\mathbb Z)\), whose eigenvalues are all some real positive numbers. This is then use to solve the CCE of the Anosov flow \(\mathcal F\) on the hyperbolic torus \({\mathbb T_A}^{n+1}\) obtained from \(A\) by suspension. Therefore some other geometrical objects associated to \(A\) and \(\mathcal F\), like invariant distributions and the leafwise cohomology are computed here.